Final answer:
The solution to the initial value problem y'' + 2y' + y = f(t), with initial conditions y(0) = 0 and y'(0) = 0, involves finding the complementary solution and, if f(t) is known, finding the particular solution to form the complete solution.
Step-by-step explanation:
The initial value problem provided is a homogeneous linear differential equation with constant coefficients, and it can be solved using the standard methods for such equations.
The known initial conditions are that y(0) = 0 and y'(0) = 0. To find the solution to the differential equation y'' + 2y' + y = f(t), we first find the complementary solution by solving the characteristic equation associated with the homogeneous part of the differential equation, which is r^2 + 2r + 1 = 0. This equation has a repeated root of r = -1. Therefore, the complementary solution will have the form y_c(t) = (C_1 + C_2t)e^{-t}. To find the particular solution, y_p(t), we need to know the form of f(t). Since f(t) is not provided, the particular solution cannot be determined.
Applying the initial conditions, we set y(0) = 0 and y'(0) = 0 to solve for C_1 and C_2. This gives us the specific solution to the initial value problem once the form of f(t) is known.
If the function f(t) were to be provided, we could use the method of undetermined coefficients or the variation of parameters to find y_p(t) and piece together the full solution as y(t) = y_c(t) + y_p(t).